Let A be an abelian category having enough projective objects and enough injective objects. We prove that if A admits an additive generating object, then the extension dimension and the weak resolution dimension of A are identical, and they are at most the representation dimension of A minus two. By using it, for a right Morita ring Λ, we establish the relation between the extension dimension of the category mod Λ of finitely generated right Λ-modules and the representation dimension as well as the right global dimension of Λ. In particular, we give an upper bound for the extension dimension of mod Λ in terms of the projective dimension of certain class of simple right Λ-modules and the radical layer length of Λ. In addition, we investigate the behavior of the extension dimension under some ring extensions and recollements.
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